What does the Qur'an say about Christians?

Answer:

In the Qur'an, Christians are often referred to as among the "People of the Book," i.e. people who have received and believed in previous revelation from God's prophets. There are verses that highlight the commonalities between Christians and Muslims, and other verses that warn Christians against sliding towards polytheism in their worship of Jesus Christ.
Commonalities:

"Surely those who believe, and those who are Jews, and the Christians, and the Sabians -- whoever believes in God and the Last Day and does good, they shall have their reward from their Lord. And there will be no fear for them, nor shall they grieve" (2:62, 5:69, and many other verses).

"...and nearest among them in love to the believers will you find those who say, 'We are Christians,' because amongst these are men devoted to learning and men who have renounced the world, and they are not arrogant" (5:82).

"O you who believe! Be helpers of God -- as Jesus the son of Mary said to the Disciples, 'Who will be my helpers in (the work of) God?' Said the disciples, 'We are God's helpers!' Then a portion of the Children of Israel believed, and a portion disbelieved. But We gave power to those who believed, against their enemies, and they became the ones that prevailed" (61:14).

Warnings:

"If only they [i.e. Christians] had stood fast by the Law, the Gospel, and all the revelation that was sent to them from their Lord, they would have enjoyed happiness from every side. There is from among them a party on the right course, but many of them follow a course that is evil" (5:66).

"Oh People of the Book! Commit no excesses in your religion, nor say of God anything but the truth. Christ Jesus, the son of Mary, was (no more than) a messenger of God, and His Word which He bestowed on Mary, and a spirit proceeding from Him. So believe in God and His messengers. Say not, 'Trinity.' Desist! It will be better for you, for God is One God, Glory be to Him! (Far exalted is He) above having a son. To Him belong all things in the heavens and on earth. And enough is God as a Disposer of affairs" (4:171).

"The Jews call 'Uzair a son of God, and the Christians call Christ the son of God. That is but a saying from their mouth; (in this) they but imitate what the unbelievers of old used to say. God's curse be on them; how they are deluded away from the Truth! They take their priests and their anchorites to be their lords in derogation of God, and (they take as their Lord) Christ the son of Mary. Yet they were commanded to worship but One God: there is no god but He. Praise and glory to Him! (Far is He) from having the partners they associate (with Him)" (9:30-31).